Problem 3: Flush Royalty
Background:
A "Royal Flush" in poker is a hand consisting of the highest ranking
cards in a given suit. Standard poker is played with hands of 5 cards
taken from a deck of 52 cards, 13 in each of four suits (clubs, diamonds,
hearts, and spades). Of all possible hands of 5 cards, only
4 are royal flushes.
However things are a little different on the planet Pokezermon.
For one thing it is a big planet, a little farther from their sun than
is the case on our familiar Earth.
It is
lost in the mists of antiquity how the customary Pokezermon card decket
came to have cards numbered 1 to 30 in each of 6 suoits (bats, ballfields,
bases, elbows, lungs, and shovels),
Perhaps it is no coicidence that the orbit and rotation rate
of Pokezermon are such that they have 180 weoks in a year.
It may be that the average number of fingets on a Pokezerian's flap
has something to do with the size of a flap of cards dealt in their
popular game, Pokeasy. However it may have evolved, the flap size
for Pokeasy is 21. Naturally, the most desirable flap in Pokeasy
is a "Flushing Royal" consisting of the 21 cards numbered 10 thru 30
all in a given one of the 6 suoits.
Well, you can imagine the Pokoid games played on other planets. On
a given planet they
use a deckel of n cards consisting of k suitems. Each suitem is of the same
size so that k divides n evenly. A dealt appendage consists of h cards,
where h < n/k. Naturally there are k very desirable "Flusheroy" appendages
each consisting of the top h cards in one of the suitems.
Input:
In this problem you produce the exact probability of a Flusheroy appendage
given n, k, and h. The input consists of a sequence of lines each containing
the three numbers characterizing the Pokoid game for a particular planet.
The deckel never has more than 999 cards so that
n is less than 1000.
Output:
For each input line, print on a single output line the probability of a
flusheroy appendage
for a Pokoid game played with a deckel of n cards, with k suitems,
and dealt appendages of h cards, where h < n/k, and k divides n evenly.
Display the probability as a rational fraction reduced to lowest terms
(no common factor in numerator and denominator). The input values
will be such that the denominator of the probability has no more
than 30 digits.
One formula for the probability is
P = number-of-suitems / number-of-possible-appendages, i.e.,
P = k / [ n!/( h! * (n - h)! ) ].
Sample Input:
52 4 5
180 6 21
Sample Output:
1/649740
1/222402674104879045094974800